On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

Abstract

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup e tL  etL. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers L N  LN of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L L. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.